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A383709 Number of integer partitions of n with distinct multiplicities (Wilf) and distinct 0-appended differences.

%I A383709 #6 May 16 2025 22:59:23
%S A383709 1,1,2,1,2,2,3,4,4,4,5,6,5,7,8,6,8,9,9,10,9,10,12,12,11,12,14,13,14,
%T A383709 15,14,16,16,16,18,17,17,19,20,19,19,21,21,22,22,21,24,24,23,25,25,25,
%U A383709 26,27,27,27,28,28,30,30,28,31,32,31,32,32,33,34,34,34
%N A383709 Number of integer partitions of n with distinct multiplicities (Wilf) and distinct 0-appended differences.
%C A383709 Integer partitions with distinct multiplicities are called Wilf partitions.
%F A383709 Ranked by A130091 /\ A325367
%e A383709 The a(1) = 1 through a(8) = 4 partitions:
%e A383709   (1)  (2)    (3)  (4)    (5)      (6)      (7)      (8)
%e A383709        (1,1)       (2,2)  (3,1,1)  (3,3)    (3,2,2)  (4,4)
%e A383709                                    (4,1,1)  (3,3,1)  (3,3,2)
%e A383709                                             (5,1,1)  (6,1,1)
%t A383709 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#]&&UnsameQ@@Differences[Append[#,0]]&]],{n,0,30}]
%Y A383709 For just distinct multiplicities we have A098859, ranks A130091, conjugate A383512.
%Y A383709 For just distinct 0-appended differences we have A325324, ranks A325367.
%Y A383709 For positive differences we have A383507, ranks A383532.
%Y A383709 These partitions are ranked by A383712.
%Y A383709 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383709 A239455 counts Look-and-Say partitions, complement A351293.
%Y A383709 A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
%Y A383709 A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
%Y A383709 A383534 gives 0-prepended differences by rank, see A325351.
%Y A383709 Cf. A047966, A320348, A325325, A325349, A325368, A325388, A381431, A383506.
%K A383709 nonn
%O A383709 0,3
%A A383709 _Gus Wiseman_, May 15 2025